On the Local structure theorem and equivariant geometry of cotangent bundles

Details On the Local structure theorem and equivariant geometry of cotangent bundles On the Local structure theorem and equivariant geometry of cotangent bundles

2010-01-09

arxiv.org/abs/1001.1421v3

Mathematics

Algebraic Geometry

27 pages, new version, cosmetic changes, a new geometric proof of the description of the normalized moment map is added (Secti

Scientific paper

Let $G$ be a connected reductive group acting on an irreducible normal algebraic variety $X$. We give a slightly improved version of local structure theorems obtained by F.Knop and D.A.Timashev that describe an action of some parabolic subgroup of $G$ on an open subset of $X$. We also extend various results of E.B.Vinberg and D.A.Timashev on the set of horospheres in $X$. We construct a family of nongeneric horospheres in $X$ and a variety $\Hor$ parameterizing this family, such that there is a rational $G$-equivariant symplectic covering of cotangent vector bundles $T^*\Hor \dashrightarrow T^*X$. As an application we get a description of the image of the moment map of $T^*X$ obtained by F.Knop by means of geometric methods that do not involve differential operators.

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